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\title{\huge \bf The Intersection of Two Cylinders \footnote{This file is from the 3D-XploreMath project. \hfil\break Please see http://vmm.math.uci.edu/3D-XplorMath/index.html or http://3d-xplormath.org/}}
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\LARGE


The image shows the space curve defined implicitly 
as the intersection of the two cylinders:
 \[ y^2 + z^2  =  f\!f \]
and 
 \[ (\cos(aa) x + \sin(aa)y)^2 + (z-cc)^2 = gg .\]
These two cylinders are made visible by displaying a random set of dots on each of them.
In the Action Menu one can choose to put more random dots on the boundary of the intersection
of the two solid cylinders.

In the default settings the two cylinders touch and the
default morph rotates one of them by changing $aa$. 

  We find it interesting to change the radius of the 
smaller cylinder while the cylinders keep touching:
morph $gg$ up to $f\!f$ while keeping $dd=0$, since we 
compute (behind the user)
\[ cc = \sqrt{f\!f} - \sqrt{gg} + dd .\]
At $gg = f\!f$ the intersection curve degenerates into
two ellipses (for each $aa$).

The distance between the tangent planes of the two cylinders (at their common normal) is $dd$.

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